Write out the lp formulation for given problem - draw the

Question

Chapter-Network Modeling

MULTIPLE CHOICE

1. Almost all network problems can be viewed as special cases of the
a. transshipment problem.
b. shortest path problem.
c. maximal flow problem.
d. minimal spanning tree problem.

2. The arcs in a network indicate all of the following except?
a. routes
b. paths
c. constraints
d. connections

3. A factory which ships items through the network would be represented by which type of node?
a. demand
b. supply
c. random
d. decision

4. A node which can both send to and receive from other nodes is a
a. demand node.
b. supply node.
c. random node.
d. transshipment node.

5. Demand quantities for demand nodes in a transshipment problem are customarily indicated by
a. positive numbers.
b. negative numbers.
c. imaginary numbers.
d. either positive or negative numbers.

6. Decision variables in network flow problems are represented by
a. nodes.
b. arcs.
c. demands.
d. supplies.

7. The number of constraints in network flow problems is determined by the number of
a. nodes.
b. arcs.
c. demands.
d. supplies.

8. How many constraints are there in a transshipment problem which has n nodes and m arcs?
a. n
b. m
c. n + m
d. m - n

9. In a transshipment problem, which of the following statements is a correct representation of the balance-of-flow rule if Total Supply < Total Demand?
a. Inflow - Outflow ≥ Supply or Demand
b. Inflow + Outflow ≥ Supply or Demand
c. Inflow - Outflow ≤ Supply or Demand
d. Inflow + Outflow ≤ Supply or Demand

10. Supply quantities for supply nodes in a transshipment problem are customarily indicated by
a. positive numbers.
b. negative numbers.
c. imaginary numbers.
d. either positive or negative numbers.

11. What is the correct constraint for node 2 in the following diagram?

a. X₁₂ + X₂₃ = 100
b. X₁₂ - X₂₃ ≤ 100
c. -X₁₂ + X₂₃ ≥ -100
d. X₁₂ - X₂₃ ≤ 100

12. The constraint X₁₃ + X₂₃ - X₃₄ ≥ 50 indicates that
a. 50 units are required at node 3.
b. 50 units will be shipped from node 3.
c. 50 units will be shipped in from node 1.
d. 50 units must pass through node 3.

13. Which balance of flow rule should be applied at each node in a network flow problem when Total Supply > Total Demand?
a. Inflow - Outflow ≤ Supply or Demand
b. Inflow - Outflow ≥ Supply or Demand
c. Inflow - Outflow = Supply or Demand
d. Inflow - Supply ≥ Outflow or Demand

14. What formula would be entered in cell G18 in this Excel model?

	A	B	C	D	E	F	G	H	I	J	K	L
1
2
3
4												Supply/
5		Ship	From		To		Unit Cost		Nodes		Net Flow	Demand
6		55	1	LAV	2	PHO	60		1	LAV	-100	-100
7		45	1	LAV	4	REN	120		2	PHO	50	50
8		5	2	PHO	3	LAX	160		3	LAX	30	30
9		0	3	LAX	5	SAN	70		4	REN	45	45
10		25	5	SAN	3	LAX	90		5	SAN	90	90
11		0	5	SAN	4	REN	70		6	DEN	35	35
12		0	5	SAN	6	DEN	90		7	SLC	-150	-150
13		0	6	DEN	5	SAN	50
14		0	7	SLC	4	REN	190
15		115	7	SLC	5	SAN	90
16		35	7	SLC	6	DEN	100
17
18						Total	25600

a. SUMPRODUCT(K6:K12,L6:L12)
b. SUMPRODUCT(B6:B16,G6:G16)
c. SUMPRODUCT(G6:G16,K6:K12)
d. SUMPRODUCT(B6:G16,L6:L12)

15. How could a network be modified if demand exceeds supply?
a. add extra supply arcs
b. remove the extra demand arcs
c. add a dummy supply
d. add a dummy demand

16. What is the interpretation of units "shipped" along arcs from dummy supply nodes to demand nodes?
a. Indicates unmet demand at demand nodes
b. Indicates unmet supply at demand nodes
c. Indicates unmet demand at supply nodes
d. Indicates unmet supply at supply nodes

17. Consider the equipment replacement problem presented in the chapter. Recall that in the network model formulation of this problem a node represents a year when the equipment was purchased. An arc from node i to node j indicates that the equipment purchased in year i can be replaced at the beginning of year j. How could the network model below be modified to depict an equipment purchase in year 4 and operating costs only through the remainder of the planning window?

a. Modify the cost on arc 4-5 to account for only operating costs.
b. Add a second arc 4-5 to represent just the operating costs.
c. Add a dummy node, 6, so that arc 4-6 represents just the operating costs.
d. Add a dummy node, 6, so that arc 4-5 represents operating costs and 5-6 represents new equipment purchase.

18. The street intersections in a city road network represent
a. nodes.
b. arcs.
c. resources.
d. expenses.

19. The right hand side value for the starting node in a shortest path problem has a value of
a. -1
b. 0
c. 1
d. 2

20. The right hand side value for the ending node in a shortest path problem has a value of
a. -1
b. 0
c. 1
d. 2

21. What is the constraint for node 2 in the following shortest path problem?

a. -X₁₂ - X₁₃ = 0
b. -X₁₂ - X₂₄ = 1
c. X₁₂ + X₁₃ = 0
d. -X₁₂ + X₂₄ = 0

22. An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. Consider arc 2-4. The per unit shipping cost of crude B from source 2 (node 2) to refinery 2 (node 4) is $11 and the yield is 85 percent. The following network representation depicts this problem. What is the balance of flow constraint for node 3 (Refinery 1)?

a. X₁₃ + X₂₃ - .95 X₃₅ - .90 X₃₆ - .90 X₃₇ = 0
b. .80 X₁₃ + .95 X₂₃ - X₃₅ - X₃₆ - X₃₇ = 0
c. .80 X₁₃ + .95 X₂₃ - .90 X₃₆ - .90 X₃₇ ≥ 0
d. X₁₃ + X₂₃ - X₃₅ - X₃₆ - X₃₇ ≥ 0

23. An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. Consider arc 2-4. The per unit shipping cost of crude B from source 2 (node 2) to refinery 2 (node 4) is $11 and the yield is 85 percent. The following flowchart depicts this problem. What is the balance of flow constraint for node 7 (Diesel)?

a. X₃₅ + X₃₆ + X₃₇ = 75
b. X₃₇ + X₄₇ ≥ 75
c. .90 X₃₇ + .95 X₄₇ = 75
d. X₃₇ + X₄₇ -X₃₆ - X₃₅ - X₄₅ - X₄₆ ≥ 75

24. A network flow problem that allows gains or losses along the arcs is called a
a. non-constant network flow model.
b. non-directional, shortest path model.
c. generalized network flow model.
d. transshipment model with linear side constraints.

25. What is the objective function for the following shortest path problem?

a. -X₁₂ - X₁₃ = 0
b. MIN -50 X₁₂ - 200 X₁₃ + 100 X₂₄ + 35 X₃₄
c. MIN 50 X₁₂ + 200 X₁₃ + 100 X₂₄ + 35 X₃₄
d. MAX -50 X₁₂ - 200 X₁₃ + 100 X₂₄ + 35 X₃₄

26. Which formula should be used to determine the Net Flow values in cell K6 in the following spreadsheet model?

	A	B	C	D	E	F	G	H	I	J	K	L
1
2
3
4												Supply/
5		Ship	From		To		Unit Cost		Nodes		Net Flow	Demand
6		55	1	LAV	2	PHO	60		1	LAV	-100	-100
7		45	1	LAV	4	REN	120		2	PHO	50	50
8		5	2	PHO	3	LAX	160		3	LAX	30	30
9		0	3	LAX	5	SAN	70		4	REN	45	45
10		25	5	SAN	3	LAX	90		5	SAN	90	90
11		0	5	SAN	4	REN	70		6	DEN	35	35
12		0	5	SAN	6	DEN	90		7	SLC	-150	-150
13		0	6	DEN	5	SAN	50
14		0	7	SLC	4	REN	190
15		115	7	SLC	5	SAN	90
16		35	7	SLC	6	DEN	100
17
18						Total	25600

a. SUMIF($C$6:$C$16,I6,$B$6:$B$16)-SUMIF($E$6:$E$16,I6,$B$6:$B$16)
b. SUMIF($I$6:$I$12,B6,$B$6:$B$16)-SUMIF($I$6:$I$12,I6,$B$6:$B$16)
c. SUMIF($E$6:$E$16,I6,$B$6:$B$16)-SUMIF($C$6:$C$16,I6,$B$6:$B$16)
d. SUMPRODUCT(B6:B16,G6:G16)

27. Which property of network flow models guarantees integer solutions?
a. linear constraints and balance of flow equation format
b. linear objective function coefficients
c. integer objective function coefficients
d. integer constraint RHS values and balance of flow equation format

28. In generalized network flow problems
a. solutions may not be integer values.
b. flows along arcs may increase or decrease.
c. it can be difficult to tell if total supply is adequate to meet total demand.
d. all of these.

29. What happens to the solution of a network flow model if side constraints are added that do not obey the balance of flow rules?
a. The model solution is not guaranteed to be integer.
b. The model solution will more accurately reflect reality.
c. The model solution will be integer but more accurate.
d. The model solution is not guaranteed to be feasible.

30. Consider modeling a warehouse with three in-flow arcs and three outflow arcs. The warehouse node is a transshipment node but has a capacity of 100. How would one modify the network model to avoid adding a side constraint that limits either the sum of in-flows or the sum of the out-flows to 100?
a. Place a limit of 34 on each in-flow arc.
b. Add a side constraint limiting the out-flow arcs sum to 100.
c. Separate the warehouse node into two nodes, connected by a single arc, with capacity of 100.
d. It cannot be accomplished, a side constraint must be added.

31. The equipment replacement problem is an example of which network problem?
a. transportation problem.
b. shortest path problem.
c. maximal flow problem.
d. minimal spanning tree problem.

32. If a side constraint for a network flow model cannot be avoided, and non-integer solutions result, how can the solution be expressed as an integer solution?
a. Force all the arc flow decision variables to be integer.
b. Round off all the non-integer arc flow decision variables.
c. Increase the supply until the solutions are all integer using a dummy supply node.
d. Increase the demand until the solutions are all integer using a dummy demand node.

33. A maximal flow problem differs from other network models in which way?
a. arcs are two directional
b. multiple supply nodes are used
c. arcs have limited capacity
d. arcs have unlimited capacity

34. Maximal flow problems are converted to transshipment problems by
a. connecting the supply and demand nodes with a return arc
b. adding extra supply nodes
c. adding supply limits on the supply nodes
d. requiring integer solutions

35. What is the objective function in the following maximal flow problem?

a. MIN X₄₁
b. MAX X₁₂ + X₁₃
c. MAX X₁₄
d. MAX X₄₁

36. What is the constraint for node 2 in the following maximal flow problem?

a. X₁₂ - X₂₃ - X₂₄ = 0
b. X₁₂ + X₂₃ + X₂₄ = 0
c. X₁₂ ≤ 4
d. X₁₂ + X₁₃ - X₂₃ = 0

37. What is missing from transportation problems compared to transshipment problems?
a. arcs
b. demand nodes
c. transshipment nodes
d. supply nodes

38. Which method is preferred for solving fully connected transportation problems?
a. linear programming
b. network flow methods
c. trial and error
d. simulation

39. When might a network flow model for a transportation/assignment problem be preferable to a matrix form for the problem?
a. When an integer solution is required.
b. When the problem is large and not fully connected.
c. When the problem is large and fully connected.
d. When supply exceeds demand.

40. Which method is preferred for solving minimal spanning tree problems?
a. linear programming
b. transshipment models
c. simulation
d. manual algorithms

41. How many arcs are required to make a spanning tree in a network with n nodes and m arcs?
a. n
b. n - 1
c. m
d. m - 1

42. The minimal spanning tree solution algorithm works by defining a subnetwork and
a. adding the least expensive arc which connects any node in the current subnetwork to any node not in the current subnetwork.
b. adding the most expensive arc which connects any node in the current subnetwork to any node not in the current subnetwork.
c. adding the least expensive arc which connects unconnected nodes in the current subnetwork.
d. adding the least expensive arc which connects the most recently added node in the current subnetwork to the closest node not in the current subnetwork.

PROBLEM

43. Draw the network representation of the following network flow problem.

MIN: 5 X₁₂ + 3 X₁₃ + 2 X₁₄ + 3 X₂₄ + 2 X₃₄
Subject to: -X₁₂ - X₁₃ - X₁₄ = -10
                  X₁₂ - X₂₄ = 2
                  X₁₃ - X₃₄ = 3
                  X₁₄ + X₂₄ + X₃₄ = 5
                  X_ij ≥ 0 for all i and j

44. A company wants to determine the optimal replacement policy for its delivery truck. New trucks cost $30,000. The company does not keep trucks longer than 2 years and has estimated the annual operating costs and trade-in values for trucks during each of the 2 years as:

	Age in years
	0-1	1-2
Operating Cost	$15,000	$16,500
Trade-in Value	$20,000	$16,000

Draw the network representation of this problem.

45. A company wants to determine the optimal replacement policy for its photocopier. The company does not keep photocopiers longer than 4 years. The company has estimated the annual costs for photocopiers during each of the 4 years and developed the following network representation of the problem.

Write out the LP formulation for this problem.

46. A company needs to ship 100 units from Roanoke to Washington at the lowest possible cost. The costs associated with shipping between the cities are:

	To
From	Lexington	Washington	Charlottesville
Roanoke	50	-	80
Lexington	-	50	40
Charlottesville	-	30

Draw the network representation of this problem.

47. A company needs to ship 100 units from Seattle to Denver at the lowest possible cost. The costs associated with shipping between the cities are listed below. Also, the decision variable associated with each pair of cities is shown next to the cost.

	To
From	Portland	Spokane	Salt Lake City	Denver
Seattle	100 (X12)	500 (X13)	600 (X14)	-
Portland	-	350 (X23)	300 (X24)	-
Spokane	-	-	250 (X34)	200 (X35)
Salt Lake City	-	-	-	200 (X45)

Write out the LP formulation for this problem.

48. A company needs to ship 100 units from Seattle to Denver at the lowest possible cost. The costs associated with shipping between the cities are:

	To
From	Portland	Spokane	Salt Lake City	Denver
Seattle	100	500	600	-
Portland	-	350	300	-
Spokane	-	-	250	200
Salt Lake City	-	-	-	200

What values should go into cells G6:L13 in the following Excel spreadsheet?

	A	B	C	D	E	F	G	H	I	J	K	L
1
2
3
4												Supply/
5		Ship	From		To		Unit Cost		Nodes		Net Flow	Demand
6			1	SEA	2	POR			1	SEA
7			1	SEA	3	SPO			2	POR
8			1	SEA	4	SLC			3	SPO
9			2	POR	3	SPO			4	SLC
10			2	POR	4	SLC			5	DEN
11			3	SPO	4	SLC
12			3	SPO	5	DEN
13			4	SLC	5	DEN
14
15				Total cost

49. A company needs to ship 100 units from Seattle to Denver at the lowest possible cost. The costs associated with shipping between the cities are:

	To
From	Portland	Spokane	Salt Lake City	Denver
Seattle	100	500	600	-
Portland	-	350	300	-
Spokane	-	-	250	200
Salt Lake City	-	-	-	200

What values would you enter in the Risk Solver Platform (RSP) task pane for the following Excel spreadsheet?

Objective Cell:

Variables Cells:

Constraints Cells:

	A	B	C	D	E	F	G	H	I	J	K	L
1
2
3
4												Supply/
5		Ship	From		To		Unit Cost		Nodes		Net Flow	Demand
6			1	SEA	2	POR	100		1	SEA	-100	-100
7			1	SEA	3	SPO	500		2	POR	0	0
8			1	SEA	4	SLC	600		3	SPO	0	0
9			2	POR	3	SPO	350		4	SLC	0	0
10			2	POR	4	SLC	300		5	DEN	100	100
11			3	SPO	4	SLC	250
12			3	SPO	5	DEN	200
13			4	SLC	5	DEN	200
14
15				Total cost

50. A trucking company wants to find the quickest route from Seattle to Denver. What values should be placed in cells L6:L10 of the following Excel spreadsheet?

	A	B	C	D	E	F	G	H	I	J	K	L
1
2
3
4		Select					Driving					Supply/
5		Route	From		To		Time		Nodes		Net Flow	Demand
6		0	1	SEA	2	POR	3		1	SEA	-1
7		0	1	SEA	3	SPO	4		2	POR	0
8		1	1	SEA	4	SLC	12		3	SPO	0
9		0	1	SEA	5	DEN	18		4	SLC	0
10		0	2	POR	3	SPO	9		5	DEN	1
11		0	2	POR	4	SLC	12
12		0	2	POR	5	DEN	16
13		0	3	SPO	4	SLC	10
14		0	3	SPO	5	DEN	15
15		1	4	SLC	5	DEN	5
16
17			Total Driving Time				17

51. An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. The following network representation depicts this problem.

Write out the LP formulation for this problem.

52. An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. The following Excel spreadsheet shows this problem. What formula should be entered in cell E6 (and copied to cells E7:E15) in this spreadsheet?

	A	B	C	D	E	F	G	H	I	J	K	L	M
1
2
3
4								Unit				Net	Supply/
5	Flow from Node			Yield	Flow into Node			Cost		Nodes		Flow	Demand
6		1	Crude A	0.90		3	Refinery 1	15		1	Crude A		-120
7		1	Crude A	0.85		4	Refinery 2	13		2	Crude B		-60
8		2	Crude B	0.80		3	Refinery 1	9		3	Refinery 1		0
9		2	Crude B	0.85		4	Refinery 2	11		4	Refinery 2		0
10		3	Refinery 1	0.95		5	Lube Oil	4		5	Lube Oil		75
11		3	Refinery 1	0.90		6	Gasoline	7		6	Gasoline		50
12		3	Refinery 1	0.90		7	Diesel	8		7	Diesel		25
13		4	Refinery 2	0.90		5	Lube Oil	3
14		4	Refinery 2	0.95		6	Gasoline	9
15		4	Refinery 2	0.95		7	Diesel	6
16
17							Total cost

53. An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. The following Excel spreadsheet shows this problem.

What values would you enter in the Risk Solver Platform (RSP) task pane for the following Excel spreadsheet?

Objective Cell:

Variables Cells:

Constraints Cells:

	A	B	C	D	E	F	G	H	I	J	K	L	M
1
2
3
4								Unit				Net	Supply/
5	Flow from Node			Yield	Flow into Node			Cost		Nodes		Flow	Demand
6		1	Crude A	0.90		3	Refinery 1	15		1	Crude A		-120
7		1	Crude A	0.85		4	Refinery 2	13		2	Crude B		-60
8		2	Crude B	0.80		3	Refinery 1	9		3	Refinery 1		0
9		2	Crude B	0.85		4	Refinery 2	11		4	Refinery 2		0
10		3	Refinery 1	0.95		5	Lube Oil	4		5	Lube Oil		75
11		3	Refinery 1	0.90		6	Gasoline	7		6	Gasoline		50
12		3	Refinery 1	0.90		7	Diesel	8		7	Diesel		25
13		4	Refinery 2	0.90		5	Lube Oil	3
14		4	Refinery 2	0.95		6	Gasoline	9
15		4	Refinery 2	0.95		7	Diesel	6
16
17							Total cost

54. Clifton Distributing has three plants and four distribution centers. The plants, their supply, the distribution centers, their demands, and the distance between each location is summarized in the following table:

Distance	Center 1	Center 2	Center 3	Center 4	Supply
Plant A	45	60	53	75	500
Plant B	81	27	49	62	700
Plant C	55	40	35	60	650
Demand	350	325	400	375

Draw the transportation network for Clifton's distribution problem.

55. The following network depicts a transportation/distribution problem for Clifton Distributing. Formulate the LP for Clifton assuming they wish to minimize the total product-miles incurred.

56. Clifton Distributing has three plants and four distribution centers. The plants, their supply, the distribution centers, their demands, and the distance between each location is summarized in the following table:

Distance	Center 1	Center 2	Center 3	Center 4	Supply
Plant A	45	60	53	75	500
Plant B	81	27	49	62	700
Plant C	55	40	35	60	650
Demand	350	325	400	375

Draw the balanced transportation network for Clifton's distribution problem.

57. The following network depicts a balanced transportation/distribution problem for Clifton Distributing. Formulate the LP for Clifton assuming they wish to minimize the total product-miles incurred.

58. Joe Fix plans the repair schedules each day for the Freeway Airline. Joe has 3 planes in need of repair and 5 repair personnel at his disposal. Each plane requires a single repairperson, except plane 3, which needs 2 personnel. Anyone not assigned to maintaining an airplane works in the maintenance shop for the day (not modeled). Each repairperson has different likes and dislikes regarding the types of repairs they prefer. For each plane, Joe has pulled the expected maintenance and determined the total preference matrix for his repair personnel. The preference matrix is:

	Plane 1	Plane 2	Plane 3
Repair Person 1	11	9	21
Repair Person 2	17	7	13
Repair Person 3	9	12	17
Repair Person 4	14	8	28
Repair Person 5	12	5	12

Draw the network flow for this assignment problem assuming Joe would like to maximize the total preference in his worker-to-aircraft schedule.

59. The following network depicts an assignment/transportation problem for Joe Fix's repair scheduling problem. Formulate the LP for Joe assuming he wishes to maximize the total repairperson to plane assignment preferences.

60. Joe Fix plans the repair schedules each day for the Freeway Airline. Joe has 3 planes in need of repair and 5 repair personnel at his disposal. Each plane requires a single repairperson, except plane 3, which needs 2 personnel. Anyone not assigned to maintaining an airplane works in the maintenance shop for the day (not modeled). Each repairperson has different likes and dislikes regarding the types of repairs they prefer. For each plane, Joe has pulled the expected maintenance and determined the total preference matrix for his repair personnel. The preference matrix is:

	Plane 1	Plane 2	Plane 3
Repair Person 1	11	9	21
Repair Person 2	17	7	13
Repair Person 3	9	12	17
Repair Person 4	14	8	28
Repair Person 5	12	5	12

Draw the balanced network flow for this assignment problem assuming Joe would like to maximize the total preference in his worker-to-aircraft schedule.

61. The following network depicts a balanced assignment/transportation problem for Joe Fix's repair scheduling problem. Formulate the LP for Joe assuming he wishes to maximize the total repairperson to plane assignment preferences.

62. A manufacturing company has a pool of 50 labor hours. A customer has requested two products, Product A and Product B, and has requested 15 and 20 of each respectively. It requires 2 hours of labor to produce Product A and 3 hours of labor to produce Product B. The company can obtain up to 50 additional hours of labor if required. In-house labor costs $25 per hour while contracted labor costs $45 per hour. Draw the network flow model that captures this problem.

63. A manufacturing company has a pool of 50 labor hours. A customer has requested two products, Product A and Product B, and has requested 15 and 20 of each respectively. It requires 2 hours of labor to produce Product A and 3 hours of labor to produce Product B. The company can obtain up to 50 additional hours of labor if required. In-house labor costs $25 per hour while contracted labor costs $45 per hour. The following network flow model captures this problem.

Write out the LP formulation for this problem.

64. A company wants to manage its distribution network which is depicted below. Identify the supply, demand and transshipment nodes in this problem.

65. Draw the network and indicate how many units are flowing along each arc based on the following Risk Solver Platform (RSP) solution.

Units					Unit				Net	Supply/
of Flow	From		To		Cost		Nodes		Flow	Demand
5	1	A	2	B	20		1	A	-40	-40
35	1	A	3	C	15		2	B	5	5
0	2	B	4	D	30		3	C	5	5
25	3	C	4	D	10		4	D	10	10
5	3	C	5	E	25		5	E	5	5
15	4	D	6	F	10		6	F	15	15
0	5	E	6	F	30
				Total	1150

66. A railroad needs to move the maximum amount of material through its rail network. Formulate the LP model to determine this maximum amount based on the following network diagram.

67. Draw the network and solution for the maximal flow problem represented by the following Excel spreadsheet.

Units					Upper				Net	Supply/
of Flow	From		To		Bound		Nodes		Flow	Demand
4	1	A	2	B	4		1	A	0	0
8	1	A	3	C	8		2	B	0	0
4	2	B	4	D	6		3	C	0	0
0	2	B	5	E	2		4	D	0	0
4	3	C	4	D	4		5	E	0	0
4	3	C	5	E	5
8	4	D	5	E	9
12	5	E	1	A	999
12	Maximal flow

68. Draw the network representation of this LP model. What type of problem is it?

MAX	X41
Subject to:	X41 - X12 - X13 = 0
	X12 - X24 = 0
	X13 - X34 = 0
	X24 + X34 - X41 = 0
	0 £ X12 £ 5,
	0 £ X13 £ 4,
	0 £ X24 £ 3,
	0 £ X34 £ 2,
	0 £ X41 £ ¥

69. Solve the following minimal spanning tree problem starting at node 1.

70. Solve the following minimal spanning tree problem starting at node 1.

71. Solve the following minimal spanning tree problem starting at node 1.

72. Project 5.1 - Recruit Training

Course	Cost Per Student	Min. Num. of Trainees	Max. Num. of Trainees
A	25	15	40
B	55	10	50
C	30	15	50
D	10	15	50
E	20	10	50
F	15	10	50

You are a military training analyst in charge of initial training for the XXX career field and must decide how to best train the new recruits to satisfy the requirements for skilled recruits. There are six different courses (A, B, C, D, E, F) used for training in the XXX career field and four different sequences of courses that can be taken to achieve the required skill level. These sequences are A-E, B, C-F, and A-D-F. The table below provides information on the six courses.

There are 100 recruits available for training and a demand for 100 skilled recruits. Assume all recruits pass each course and that you are trying to put students in classes in order to minimize the total cost of training. Assume non-integer solutions are acceptable. Further, assume each course will be held.

a. Draw a network flow diagram describing the problem.

b. Formulate the associated network flow linear program.

c. Implement a spreadsheet model and use Risk Solver Platform (RSP) to obtain a solution to the problem. Use your model to answer the following questions.

What is the expected student load for each course?
Should any course be expanded?
Should any course or sequence be considered for elimination?

Next, assume that not all students pass each course. In fact only 90% of the students pass courses A, E, and F and only 95% of the students pass courses B, C, and D. Each course is considered independent. The requirement for 100 skilled recruits remains. Your job is now to determine the number of recruits to place into the training program to obtain the 100 trained recruits while continuing to minimize the total cost of training.

d. Re-draw the network flow diagram describing the problem to accommodate the above changes.

e. Formulate the associated generalized network flow linear program.

f. Implement a spreadsheet model of this changed model and use Risk Solver Platform (RSP) to obtain a solution to the expanded problem. How many recruits are needed and what is the change in total training cost?

73. Project 5.2 Small Production Planning Project

(Fixed Charge Problem via Network Flow with Side Constraints)

Jack Small Enterprises runs two factories in Ohio, one in Toledo and one in Centerville. His factories produce a variety of products. Two of his product lines are polished wood clocks which he adorns with a regional theme. Naturally, clocks popular in the southwest are not as popular in the northeast, and vice versa. Each plant makes both of the clocks. These clocks are shipped to St Louis for distribution to the southeast and western states and to Pittsburg for distribution to the south and northeast.

Jack is considering streamlining his plants by removing certain production lines from certain plants. Among his options is potentially eliminating the clock production line at either the Toledo or the Centerville plant. Each plant carries a fixed operating cost for setting up the line and a unit production cost, both in terms of money and factory worker hours. This information is summarized in the table below.

		Production Cost per Clock		Clocks Produced per Hour		Available
Plant	Fixed Cost for Line	Southwest Clock	Northeast Clock	Southwest Clock	Northwest Clock	Hours per Month
Toledo	$20,000	$10	$12	5	6	500
Centerville	$24,000	$  9	$13	5.5	6.2	675

The Southwest clocks are sold for $23 each and the Northwest clocks are sold for $25 each. Demand rates used for production planning are 1875 Southwest clocks for sale out of the St Louis distribution center and 2000 Northeast clocks for sale out of the Pittsburg distribution center. Assume all these units are sold. The per clock transportation costs from plant to distribution center is given in the following table.

(cost per clock shipped)	Cost to Ship to Distribution Center
Plant	St Louis	Pittsburg
Toledo	$2	$4
Centerville	$3	$2

Develop a generalized network flow model for this problem and implement this model in solver. Use the model to answer the following questions.

a. Should any of the production lines be shut down?

b. How should worker hours be allocated to produce the clocks to meet the demand forecasts? Are there any excess hours, and if so how many?

c. What is the expected monthly profit?

d. If a plant is closed, what are the estimated monthly savings?